Friedland-Hersonsky problem for matrix algebra (Q1410711)

The aim of this article is to answer a question posed by \textit{S. Friedland} and \textit{S. Hersonsky} [in Duke Math. J. 69, No. 3, 593-614 (1993; Zbl 0799.30033)], concerning the generalization of Jørgensen's inequality to normed algebras. To state the question precisely, let \(A\in\text{GL}_n(\mathbb{C})\) and define the operator on \(M_n(\mathbb{C})\) (the set of \(n\times n\) matrices with entries in \(\mathbb{C}\)) by \(\widehat A(Z)=AZA^{-1}-Z\), where \(Z\in M_n(\mathbb{C})\). Let \(|\cdot|\) be a vector norm on \(M_n(\mathbb{C})\), the norm of the operator \(\widehat A\) is defined by \[ \|\widehat A\|=\sup_{|Z|\leq 1}|\widehat A(Z)|. \] Question: For a diagonal matrix \(A=\text{diag}(\lambda_1,\dots,\lambda_n)\in\text{GL}_n(\mathbb{C})\), does equality hold in the inequality \[ \max_{1\leq i,j\leq n}|\tfrac{\lambda_i}{\lambda_j}-1|\leq\|\widehat A\| \] if we take \(|\cdot|\) to be the spectral norm? (The inequality is a theorem of Friedland-Hersonsky.) The authors show that the answer to the above question is no when \(n\geq 3\) by constructing specific counterexamples, and also by proving some general results on conditions for which equality holds in the inequality. They also show that the answer to the question is yes if \(n=2\).